As shown in the figure, an equilateral triangle A B C is formed by joining three rods of equal lengths and D is the midpoint of A B . Coefficient of linear expansion of the material of A B is ₁ and that of A C and B C is ₂ . If the length D C remains constant for small changes in temperature, then

Step-by-step solution

For a change of temperature by t , change in length D C is D C , then D C^2= A C^2- A D^2 aligned & =l (1+ ₂ t )^2- ( l 2 (1+ ₂ t ) )^2 & =l^2 (2 ₂ t )- l^2 4 (2 ₁ t ) aligned (After neglecting terms ₂^2 t^2 and ₁^2 t^2 , being very small) As, D C=0 l^2 2 ₂ t- l^2 4 (2 ₁ t )=0 ₁=4 ₂